rear axle bent?

In looking over a 1990 Schwinn Criss-Cross (getting it ready for my son), I noticed that when I lift the rear wheel and spin it, the cogs seem to move (not rotate) and sway some. Seems to indicate the axle has a warp. Oddly, the wheel, though not quite true, isn't too bad.

Am I diagnosing this correctly?
Is there a fix? or LBS trip? or trash?

too many bikes from 1967 10s (5x2)Frejus to a Sumitomo Ti/Chorus aluminum 10s (10x2), plus one non-susp mtn bike I use as my commuter

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Freewheel wobble is normal and not indicative of anything. It happens because the axis of the freewheels thread and the hubs aren't perfectly aligned, so as the hub spins it wobbles the axis of the stationary freewheel. if there's nothing else going on, don't sweat it.

If you're concerned about the axle, diagnose it by spinning it while the wheel is stationary. The tip will wobble if it's bent.